Content Writer
Quadrants are defined as the four infinite regions of a two-dimensional cartesian system. Each region is bounded by two half-axes. Quadrants are written in Roman numerals, which are numbered from 1st to 4th . Every cartesian plane is divided into four regions. The regions are named as the first quadrant, second quadrant, third quadrant, and fourth quadrant.
- Quadrants includes both positive and negative value of the x-axis and y-axis.
- The axes are numbered in the counterclockwise direction.
- The numbering of quadrants starts from the upper right or northeast segment.
- The origin is the point of intersection of the x and y-axis in a cartesian plane.
- Surveyors use the concept of quadrants to calculate the heights of buildings and mountains.
- Quadrants are used by navigators to find the right direction.
Read More: Sin2x Formula
Key Terms: Quadrants, Cartesian System, Axes, Point of Reference, Sine, Cosine, Tangent, Trigonometric function, Abscissa, Ordinate
Quadrants
[Click Here for Sample Questions]
Quadrants are the areas which are formed when two coordinate axes of the plane intersect with each other at an angle of 90 degree. The intersection of these two lines is known as a point of reference.
- Quadrants includes two lines of intersection namely the horizontal X axis and a vertical Y axis.
- The sign of each of the quadrant (x; y) are I (+; +), II (−; +), III (−; −), and IV (+; −).
- It is easy to learn the sign of each quadrant by remembering the expression “All Science Teachers Crazy" .
- The expression can further be elaborated as: "All" functions are positive in quadrant I.
- "Science" stands for sine functions which are positive in quadrant II.
- "Teachers" stands for tangent functions which are positive in quadrant III.
- "Crazy" stands for cosine functions which are is positive in quadrant IV.
Read More:
The video below explains this:
Coordinate Geometry Detailed Video Explanation:
Types of quadrants
[Click Here for Sample Questions]
Quadrants are divided into four regions which are as follows:
First Quadrant
The first quadrant is located at the upper righthand corner of the plane. It includes a positive x axis and a positive y axis. The values of both x and y axis are positive in this quadrant. The geometric angle here lies between 0 degree to 90 degree (0°- 90°).
Read More: Inverse Trigonometric Formulas
Second Quadrant
The second quadrant lies at the left hand top side region on the cartesian plane. Here, the value of the x axis becomes negative, whereas the value of the y axis remains positive. This quadrant is denoted by 90 degrees to 180 degrees (90° - 180°).
Third Quadrant
The third quadrant is located directly below the second quadrant. In this quadrant, the values of both the x axis and the y axis are negative. The angles of this quadrant range from 180 degrees to 270 degrees (180° - 270°).
Read More: Congruence of Triangles
Fourth Quadrant
The fourth quadrant is located at the bottom right corner. In this quadrant, the value of the x axis is positive whereas the value of the y axis becomes negative here. The angle of this quadrant falls between 270 degrees to 360 degrees (270° - 360°).
Sign Convention in Quadrant
[Click Here for Sample Questions]
The xy plane is divided into four quadrants and the point in each quadrant will have a different value of x and y.
| Quadrants | Value of X - coordinate | Value of Y - coordinate |
|---|---|---|
| 1st quadrant | Becomes Positive | Becomes Positive |
| 2nd quadrant | Becomes Negative | Becomes Positive |
| 3rd quadrant | Becomes Negative | Becomes Negative |
| 4th quadrant | Becomes Positive | Becomes Negative |
Read More: Law of Sines
Plotting points on a graph
[Click Here for Sample Questions]
There are certain things we need to keep in mind before plotting a graph for a quadrant.
- The coordinates should be written in the form of (a,b) where a is the x coordinate and ‘b’ is the y coordinate.
- Abscissa is known as the sign of the x coordinate.
- It is the distance of a point from the vertical line or y-axis.
- Abscissa is measured parallel to the horizontal x-axis.
- Ordinate can be referred to as how distant a point is from the x-axis.
- It is measured in a parallel manner with respect to the y-axis lying vertically on the graph.
- The point of intersection where both the axes meet is called ‘origin’ and is denoted by (0,0).
- It implies that the value of the x-axis here is 0, and the value of the y-axis is 0.
For example:
- Suppose we want to point coordinates (5,6) on the (0,0), then we will mark five on the positive side of the x-axis, that is, the first quadrant.
- Next, we will mark six on the y-axis in the first quadrant, as both of the coordinate values are positive here.
- Then we will put a point on (0,0), which is the origin, and finally join the two points by drawing a line.
Quadrants
Read More: Tan 0 Degrees
Trigonometric Values of Quadrants
[Click Here for Sample Questions]
There are six different types of trigonometric functions. These are sine, cosine, tangent, cot, sec, and cosec. Each of them has different values in the quadrant.
- In the first quadrant and the second quadrant, the sine function has positive values in the graph.
- This function records negative values in the third and fourth quadrants.
- Cosec values are similar to that of sine, so they will have positive values and negative values like that of sine function.
- The sec function has similar values to that of cosines.
- It will have positive values in the first and fourth quadrants.
- Sec will have negative values in the second and third quadrants.
- The tangent trigonometric function has positive values in the first and third quadrants.
- Apart from this, it reads negative values in the second and fourth quadrants.
- The cost function has similar values to that of the tan function.
Read More: Trapezoids
| Trigonometric Function | 1st Quadrant | 2nd Quadrant | 3rd Quadrant | 4th Quadrant |
|---|---|---|---|---|
| Sin | +ve | +ve | -ve | -ve |
| Cos | +ve | -ve | -ve | +ve |
| Tan | +ve | -ve | +ve | -ve |
| Cot | +ve | -ve | +ve | -ve |
| Sec | +ve | -ve | -ve | +ve |
| Cosec | +ve | +ve | -ve | -ve |
Read More: Sphere Formula
Solved Examples of QuadrantsExample 1: In which quadrant do the following points lie? (A) (6, - 6) (B) (√5 - 3, - 3) (C) (√2 - 9, 4 -√3) Ans. (A) (6, - 6): Quadrant IV, because the x-coordinate is positive and y-coordinate is negative. (B) (√5 - 3, - 2): Quadrant III, because the x-coordinate and y-coordinate are both negative. (C) (√2 - 9, 4 -√3): Quadrant I, because the x-coordinate and y-coordinate are both positive. Read More: Types of Relation Example 2: Give an example of a point that lies in the second quadrant? Ans. In the second quadrant, the value of the x axis becomes negative, whereas the value of the y axis remains positive. (-3, 7) is the example of a point in the second and third quadrant. |
Uses of Quadrants
[Click Here for Sample Questions]
The various uses of quadrants are as follows:
- Quadrants was used by sailors and navigators to measure the height of the pole star and the sun while travelling.
- Surveyors also use quadrants to anticipate heights of buildings and mountains.
- Quadrants was used in war in weapons like canons, so as to provide a better aim.
- It is used navigators which helps them find the right direction.
- There are many quadrants used in astronomy for different and specific purposes.
Read More: Difference Between Area and Volume
Things to Remember
- Quadrants are the areas which are formed when two coordinate axes of the plane intersect with each other at an angle of 90 degrees.
- The point where the x-axis and the y-axis intersect is known as the origin with coordinates (0,0).
- Quadrants are divided into four regions, namely the first quadrant, second quadrant, third quadrant, and fourth quadrant.
- Both the x-axis and y-axis are positive in the first quadrant.
- In the second quadrant, the x-axis is negative, while the y-axis is positive.
- Both the x-axis and the y-axis are positive in the third quadrant.
- In the fourth quadrant, the x-axis is positive while the y-axis is negative.
- The values in each quadrant keep changing as we move anticlockwise.
Read More: Difference Between Parabola and Hyperbola
Sample Questions
Ques. Which quadrant is called the positive quadrant? (2 Marks)
Ans. The first quadrant, that is situated at the upper right corner, is called the positive quadrant. The reason behind this is the values of both the x axis and y axis are positive on the plane. They are positive coordinates. Both the x-axis and y-axis are positive in the first quadrant.
Ques. What are quadrants? How many types of quadrants are there? (3 Marks)
Ans. Quadrants are the enclosed regions formed, when two lines (one horizontal and one vertical) intersect with one another in a plane. There are 4 types of quadrants, called 1st quadrant, 2nd quadrant, 3rd quadrant and 4th quadrant. While calculations, we consider them in an anticlockwise direction.
- In first quadrant, both the x-axis and y-axis are positive in the first quadrant.
- In the second quadrant, the x-axis is negative, while the y-axis is positive.
- Both the x-axis and the y-axis are positive in the third quadrant.
- In the fourth quadrant, the x-axis is positive while the y-axis is negative.
Ques. What is the point where two lines on a plane meet, called? (2 Marks)
Ans. The point where two lines intersect is known as the origin. Its coordinates are (0,0), where first 0 denotes the value of the x coordinate and second zero refers to the y coordinate. At the origin both of them are 0.
Ques. In which quadrants, do these coordinate points take place? (4 Marks)
(A) (-2,7)
(B) (3, -5)
(C) (-4,-4)
(D) (2,6)
Ans. (A) (-2,7) lies in the second quadrant. Here, the value of the x axis becomes negative.
(B) (3, -5) lies in the fourth quadrant. It indicates that the value of the y axis is negative, while the x axis remains positive.
(C) (-4,-4) lie in the third quadrant where both of them become negative.
(D) Coordinate (2,6) lie in the first quadrant. This is because both the values are positive.
Ques. Plot the points in the graph? (4 marks)
(A) (-3,-5)
(B) (4,3)
Ans: 
The point ‘B’ denotes the coordinates (-3, -5), whereas point ‘A’ denotes the points (3,4) on the graph.
Ques. Where does the point (-5, 0) lie in the cartesian plane? Does it lie in a quadrant? (2 Marks)
Ans. Quadrants is divided into four infinite regions of a two-dimensional cartesian system. Each region is bounded by two half-axes.The point (-5, 0) lies on the horizontal axis or x-axis, which is at a distance of 5 units from the origin. It does not lie in any quadrant.
Ques. Check and answer to which quadrants do the given points lie? (4 Marks)
(A) (2, 2)
(B) (-2, -5)
(C) (-2, 9)
(D) (6, -6)
Ans. Since, the coordinates of (2, 2) are both positive, thus it lies in the 1st quadrant.
The coordinates of the point (-2, -5) are both negative, thus it lies in the 3rd quadrant.
The coordinates of the point (-2, 9) are negative on the x-axis and positive on the y-axis, thus it lies in the 2nd quadrant.
The coordinate of the point (6, -6) are positive on the x-axis and negative on the y-axis, thus it lies in the 4th quadrant.
Ques. What are the uses of quadrants? (3 marks)
Ans. The uses of quadrants are as follows:
- It is used by sailors to measure the height of the pole star while travelling.
- Architect use quadrants to anticipate heights of buildings and mountains.
- Quadrants was used in war in weapons like canons, so as to provide a better aim.
- Navigators used quadrants to find the right direction.
Ques. Identify the quadrants in which each of the following points lie? (4 Marks)
(A) (5 , 3)
(B) (4 ,-3)
(C) (-6 ,-4)
(D) (-1, 9)
Ans.(A) (5 , 3): Quadrant I, because the x-coordinate and y-coordinate are both positive.
(B) (4 ,-3): Quadrant IV, because the x-coordinate is positive and y-coordinate is negative.
(C) (-6 ,-4): Quadrant III, because the x-coordinate and y-coordinate are both negative.
(D) (-1, 9): Quadrant II, because the x-coordinate is negative and y-coordinate is positive.
Ques. In which quadrant do the following points lie? (4 Marks)
(A) (3, - 6)
(B) (√5 - 2, - 2)
(C) (√2 - 3, 2 -√3)
(D) (- 4, √7- 6)
Ans. (A) (3, - 6): Quadrant IV, because the x-coordinate is positive and y-coordinate is negative.
(B) (√5 - 2, - 2): Quadrant III, because the x-coordinate and y-coordinate are both negative.
(C) (√2 - 3, 2 -√3): Quadrant I, because the x-coordinate and y-coordinate are both positive.
(D) (- 4, √7- 6): Quadrant II, because the x-coordinate is negative and y-coordinate is positive.
Ques. Give an example of a point that lies in the second and third quadrant? (2 marks)
Ans. In the second quadrant, the value of the x axis becomes negative, whereas the value of the y axis remains positive. In the third quadrant, the values of both the x axis and the y axis are negative. The angles of this quadrant range from 180 degrees to 270 degrees (180° - 270°). (-1, 9) and (-1, -3) are examples of a point in the second and third quadrant.
Ques. Where does the point (-7, 0) lie in the cartesian plane? Does it lie in a quadrant? (2 Marks)
Ans. Quadrants is divided into four infinite regions of a two-dimensional cartesian system. Each region is bounded by two half-axes.The point (-5, 0) lies on the horizontal axis or x-axis, which is at a distance of 7 units from the origin. It does not lie in any quadrant.
Ques. Give an example of a point that lies in the first and fourth quadrant? (2 marks)
Ans. In the first quadrant, the value of the x axis and y axis are positive. The geometric angle here lies between 0 degree to 90 degree (0°- 90°). In the fourth quadrant, the values of x axis is positive whereas the value of the y axis becomes negativee. The angle of this quadrant falls between 270 degrees to 360 degrees (270° - 360°). (1, 7) and (1, -6) are examples of a point in the first and fourth quadrant.
Read More:
Comments